I am guessing it would be the genuine Fender American Std nut. But how can I be sure without pulling the nut off first? It still works some and I'd rather leave it in if I can. But I certainly need a replacement.

It's a 3/16 for sure. The adjusting t-handle ball end wrench that came with it is that size.

Actually, you may have to remove it. Using a screw extractor (found at any hardware store; ask the sales rep for any needed assistance; you may even be able to get it removed there in the store without paying), you remove the nut, and then can use bolts in the store to find the size of the thread. With that info, you can get the replacement. The whole process, if you have cooperative hardware folks and a well-stocked guitar shop (although you want to be REALLY careful; they can accidentally ruin your rod by getting the nut stuck on it), will take, at the most, two hours. Otherwise, prolly two days.